Generating functions and short recursions, with applications to the moments of quadratic forms in noncentral normal vectors

Grant H. Hillier, Raymond Kan and Xiaolu Wang

No CWP14/08, CeMMAP working papers from Centre for Microdata Methods and Practice, Institute for Fiscal Studies

Abstract:

Using generating functions, the top-order zonal polynomials that occur in much distribution theory under normality can be recursively related to other symmetric functions (power-sum and elementary symmetric functions, Ruben, Hillier, Kan, and Wang). Typically, in a recursion of this type the k-th object of interest, d_k say, is expressed in terms of all lower-order d_j's. In Hillier, Kan, and Wang we pointed out that, in the case of top-order zonal polynomials (and generalizations of them), a shorter (i.e., fixed length) recursion can be deduced. The present paper shows that the argument in generalizes to a large class of objects/generating functions. The results thus obtained are then applied to various problems involving quadratic forms in noncentral normal vectors.

JEL-codes: C16 C46 C63 (search for similar items in EconPapers)
New Economics Papers: this item is included in nep-ecm
Date: 2008-06
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